Methods and apparatus for calculating substrate model parameters and controlling lithographic processing

ABSTRACT

Metrology measurements are performed on substrates that have been subjected to lithographic processing. Model parameters are calculated by fitting the measurements to an extended high-order substrate model defined using a combination of basis functions that include an edge basis function related to a substrate edge. A radial edge basis function may be expressed in terms of distance from a substrate edge. The edge basis function may, for example, be an exponential decay function or a rational function. Lithographic processing of a subsequent substrate is controlled using the calculated high-order substrate model parameters, in combination with low-order substrate model parameters obtained by fitting inline measurements to a low order model.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is the U.S. national phase entry of PCT patentapplication no. PCT/EP2015/076432, which was filed on Nov. 12, 2015,which claims the benefit of priority of European patent application no.14197517.7, which was filed on Dec. 12, 2014, and which is incorporatedherein in its entirety by reference.

BACKGROUND

Field of the Invention

The invention relates to methods and apparatus for calculating substratemodel parameters, to methods and apparatus for controlling lithographicprocessing and to computer program products for implementing suchmethods and apparatus. The substrate model parameters can be used forexample in models for correcting errors of overlay and alignment inlithographic processing.

Related Art

A lithographic process is one in which a lithographic apparatus appliesa desired pattern onto a substrate, usually onto a target portion of thesubstrate, after which various processing chemical and/or physicalprocessing steps work through the pattern to create functional featuresof a complex product. The accurate placement of patterns on thesubstrate is a chief challenge for reducing the size of circuitcomponents and other products that may be produced by lithography. Inparticular, the challenge of measuring accurately the features on asubstrate which have already been laid down is a critical step in beingable to position successive layers of features in superpositionaccurately enough to produce working devices with a high yield.So-called overlay should, in general, be achieved within a few tens ofnanometers in today's sub-micron semiconductor devices, down to a fewnanometers in the most critical layers.

Consequently, modern lithography apparatuses involve extensivemeasurement or ‘mapping’ operations prior to the step of actuallyexposing or otherwise patterning the substrate at a target location. Inthe following discussion, the substrate will be referred to forconvenience as a “wafer”, without implying any limitation to the typesof substrate that may be processed using the invention. Advancedsubstrate models, for example alignment models, have been, and continueto be, developed to model and correct more accurately non-lineardistortions of the wafer grid that are caused by processing steps and/orby the lithographic apparatus itself. The expression wafer grid is usedto refer to a coordinate system that is formed by the (measured)alignment marks at the wafer. For example, a wafer grid is formed by thealignment marks in the scribe lanes of the wafer, that in the ideal caseform an orthogonal grid.

Alignment model parameters are calculated in order to fit an alignmentmodel to measurements of structures on substrates. Overlay and alignmenterror on production wafers as function of a position on the wafer candescribed by means of alignment models. These alignment models are usedin automatic process control (APC) systems to control lithographicprocesses to correct for overlay and alignment errors. However, it hasbeen found that even with such correction, there is still yield loss atthe edge of the wafer. To correct this yield loss using known modellingtechniques would introduce a high burden on measurement and oncomputation.

BRIEF SUMMARY OF THE INVENTION

The inventors have recognized that substrate models can be improved toaddress localized effects at the substrate edge, without undue increasein computational or measurement overhead.

The invention in a first aspect provides a method of calculating asubstrate model for use in controlling a lithographic process, themethod comprising:

-   -   defining a substrate model for representing disturbances of        features on substrate to which patterns are to be applied by the        lithographic process, the substrate model being defined as a        combination of predefined basis functions;    -   receiving measurements of structures on at least one substrate;        and    -   calculating substrate model parameters using the measurements        and the basis functions,

wherein the basis functions include at least one edge basis function forrepresenting effects related to a substrate edge.

The invention in a second aspect provides an apparatus for calculating asubstrate model for use in controlling a lithographic process, theapparatus comprising a data processing apparatus programmed to performthe steps of:

-   -   defining a substrate model for representing disturbances of        features on substrate to which patterns are to be applied by the        lithographic process, the substrate model being defined as a        combination of predefined basis functions;    -   receiving measurements of structures on at least one substrate;        and    -   calculating substrate model parameters using the measurements        and the basis functions,

wherein the basis functions include at least one edge basis function forrepresenting effects related to a substrate edge.

The invention in a third aspect provides a method of controllinglithographic processing in which patterns are applied to substrates, themethod comprising:

-   -   receiving first measurements of structures on substrates that        have been subjected to lithographic processing;    -   using said first measurements of disturbances to calculate first        substrate model parameters using the method of the first aspect        or the apparatus of the second aspect; and    -   controlling lithographic processing of a subsequent substrate        using said calculated first substrate model parameters.

The step of controlling lithographic processing of the subsequentsubstrate may comprise:

-   -   receiving second measurements of disturbances on the subsequent        substrate, the second measurements having a lower number per        substrate than the first measurements;    -   using said second measurements to calculate second substrate        model parameters of the subsequent substrate using a model        having a lower order than a model used to calculate the first        substrate model parameters; and    -   controlling lithographic processing of the subsequent substrate        using said calculated first substrate model parameters in        combination with said second substrate model parameters.

The invention in a fourth aspect provides an apparatus for controllinglithographic processing in which substrates are subjected tolithographic processing, the apparatus comprising a data processingapparatus programmed to perform the steps of:

-   -   receiving first measurements of disturbances on substrates that        have been subjected to lithographic processing;    -   using said first measurements to calculate first substrate model        parameters using the method of the first aspect or the apparatus        of the second aspect; and    -   controlling lithographic processing of a subsequent substrate        using said calculated first substrate model parameters.

The step of controlling lithographic processing of the subsequentsubstrate may comprise:

-   -   receiving second measurements of disturbances on the subsequent        substrate, the second measurements having a lower number per        substrate than the first measurements;    -   using said second measurements to calculate second substrate        model parameters of the subsequent substrate using a model        having a lower order than a model used to calculate the first        substrate model parameters; and    -   controlling lithographic processing of the subsequent substrate        using said calculated first substrate model parameters in        combination with said second substrate model parameters.

The invention in a fifth aspect provides a method of controllinglithographic processing, the method comprising:

-   -   calculating substrate model parameters using the method of the        first aspect or the apparatus of the second aspect; and    -   controlling lithographic processing using said calculated        substrate model parameters.

The invention in a sixth aspect provides an apparatus for controllinglithographic processing, the apparatus comprising a data processingapparatus programmed to perform the steps of:

-   -   calculating substrate model parameters using the method of the        first aspect or the apparatus of the second aspect; and    -   controlling lithographic processing using said calculated        substrate model parameters.

The invention in a seventh aspect provides a computer program productcomprising machine readable instructions for causing a general purposedata processing apparatus to perform the steps of the method of thefirst aspect or to implement the apparatus of the second aspect.

Further features and advantages of the invention, as well as thestructure and operation of various embodiments of the present invention,are described in detail below with reference to the accompanyingdrawings. It is noted that the invention is not limited to the specificembodiments described herein. Such embodiments are presented herein forillustrative purposes only. Additional embodiments will be apparent topersons skilled in the relevant art(s) based on the teachings containedherein.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will now be described, by way of exampleonly, with reference to the accompanying schematic drawings in which:

FIG. 1 depicts a lithographic apparatus according to an embodiment ofthe invention;

FIG. 2 shows schematically the use of the lithographic apparatus of FIG.1 together with other apparatuses forming a production facility forsemiconductor devices;

FIG. 3 illustrates schematically measurement and exposure processes inthe apparatus of FIG. 1;

FIGS. 4A to 4C illustrate alignment information being used to correctfor wafer grid distortion;

FIG. 5 illustrates examples of alignment errors and residuals for twoexample multi-wafer lots, with arrows indicating the direction ofalignment errors;

FIG. 6 illustrates examples of root causes and the effect on overlay ofwafer edge effects;

FIG. 7 illustrates graphs of radial alignment error versus distance fromthe wafer edge with fitted models having parameters calculated inaccordance with embodiments of the present invention;

FIG. 8 illustrates a method of controlling lithographic processing inaccordance with an embodiment of the present invention; and

FIG. 9 illustrates schematically data processing hardware programmableto implement the apparatuses of the embodiments of the presentinvention.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

FIG. 1 schematically depicts a lithographic apparatus LA. The apparatuscomprises:

-   -   an illumination system (illuminator) IL configured to condition        a radiation beam B (e.g. UV radiation or EUV radiation).    -   a support structure (e.g. a mask table) MT constructed to        support a patterning device (e.g. a mask) MA and connected to a        first positioner PM configured to accurately position the        patterning device in accordance with certain parameters;    -   a substrate table (e.g. a wafer table) WTa or WTb constructed to        hold a substrate (e.g. a resist coated wafer) W and connected to        a second positioner PW configured to accurately position the        substrate in accordance with certain parameters; and    -   a projection system (e.g. a refractive projection lens system)        PS configured to project a pattern imparted to the radiation        beam B by patterning device MA onto a target portion C (e.g.        comprising one or more dies) of the substrate W.

The illumination system may include various types of optical components,such as refractive, reflective, magnetic, electromagnetic, electrostaticor other types of optical components, or any combination thereof, fordirecting, shaping, or controlling radiation.

The support structure supports, i.e. bears the weight of, the patterningdevice. It holds the patterning device in a manner that depends on theorientation of the patterning device, the design of the lithographicapparatus, and other conditions, such as for example whether or not thepatterning device is held in a vacuum environment. The support structurecan use mechanical, vacuum, electrostatic or other clamping techniquesto hold the patterning device. The support structure may be a frame or atable, for example, which may be fixed or movable as required. Thesupport structure may ensure that the patterning device is at a desiredposition, for example with respect to the projection system. Any use ofthe terms “reticle” or “mask” herein may be considered synonymous withthe more general term “patterning device.”

The term “patterning device” used herein should be broadly interpretedas referring to any device that can be used to impart a radiation beamwith a pattern in its cross-section such as to create a pattern in atarget portion of the substrate.

It should be noted that the pattern imparted to the radiation beam maynot exactly correspond to the desired pattern in the target portion ofthe substrate, for example if the pattern includes phase-shiftingfeatures or so called assist features. Generally, the pattern impartedto the radiation beam will correspond to a particular functional layerin a device being created in the target portion, such as an integratedcircuit.

The patterning device may be transmissive or reflective. Examples ofpatterning devices include masks, programmable mirror arrays, andprogrammable LCD panels. Masks are well known in lithography, andinclude mask types such as binary, alternating phase-shift, andattenuated phase-shift, as well as various hybrid mask types. An exampleof a programmable mirror array employs a matrix arrangement of smallmirrors, each of which can be individually tilted so as to reflect anincoming radiation beam in different directions. The tilted mirrorsimpart a pattern in a radiation beam which is reflected by the mirrormatrix.

The term “projection system” used herein should be broadly interpretedas encompassing any type of projection system, including refractive,reflective, catadioptric, magnetic, electromagnetic and electrostaticoptical systems, or any combination thereof, as appropriate for theexposure radiation being used, or for other factors such as the use ofan immersion liquid or the use of a vacuum. Any use of the term“projection lens” herein may be considered as synonymous with the moregeneral term “projection system”.

As here depicted, the apparatus is of a transmissive type (e.g.employing a transmissive mask). Alternatively, the apparatus may be of areflective type (e.g. employing a programmable mirror array of a type asreferred to above, or employing a reflective mask).

The lithographic apparatus may be of a type having two (dual stage) ormore substrate tables (and/or two or more mask tables). In such“multiple stage” machines the additional tables may be used in parallel,or preparatory steps may be carried out on one or more tables while oneor more other tables are being used for exposure.

The lithographic apparatus may also be of a type wherein at least aportion of the substrate may be covered by a liquid having a relativelyhigh refractive index, e.g. water, so as to fill a space between theprojection system and the substrate. An immersion liquid may also beapplied to other spaces in the lithographic apparatus, for example,between the mask and the projection system. Immersion techniques arewell known in the art for increasing the numerical aperture ofprojection systems. The term “immersion” as used herein does not meanthat a structure, such as a substrate, must be submerged in liquid, butrather only means that liquid is located between the projection systemand the substrate during exposure.

Illuminator IL receives a radiation beam from a radiation source SO. Thesource and the lithographic apparatus may be separate entities, forexample when the source is an excimer laser. In such cases, the sourceis not considered to form part of the lithographic apparatus and theradiation beam is passed from the source SO to the illuminator IL withthe aid of a beam delivery system BD comprising, for example, suitabledirecting mirrors and/or a beam expander. In other cases the source maybe an integral part of the lithographic apparatus, for example when thesource is a mercury lamp. The source SO and the illuminator IL, togetherwith the beam delivery system BD if required, may be referred to as aradiation system.

The illuminator IL may comprise an adjuster AD for adjusting the angularintensity distribution of the radiation beam. Generally, at least theouter and/or inner radial extent (commonly referred to as σ-outer andσ-inner, respectively) of the intensity distribution in a pupil plane ofthe illuminator can be adjusted. In addition, the illuminator IL maycomprise various other components, such as an integrator IN and acondenser CO. The illuminator may be used to condition the radiationbeam, to have a desired uniformity and intensity distribution in itscross section.

The radiation beam B is incident on the patterning device (e.g., maskMA), which is held on the support structure (e.g., mask table MT), andis patterned by the patterning device. Having traversed the mask MA, theradiation beam B passes through the projection system PS, which focusesthe beam onto a target portion C of the substrate W. With the aid of thesecond positioner PW and position sensor IF (e.g. an interferometricdevice, linear encoder or capacitive sensor), the substrate tableWTa/WTb can be moved accurately, e.g. so as to position different targetportions C in the path of the radiation beam B. Similarly, the firstpositioner PM and another position sensor (which is not explicitlydepicted in FIG. 1) can be used to accurately position the mask MA withrespect to the path of the radiation beam B, e.g. after mechanicalretrieval from a mask library, or during a scan. In general, movement ofthe mask table MT may be realized with the aid of a long-stroke module(coarse positioning) and a short-stroke module (fine positioning), whichform part of the first positioner PM. Similarly, movement of thesubstrate table WTa/WTb may be realized using a long-stroke module and ashort-stroke module, which form part of the second positioner PW. In thecase of a stepper (as opposed to a scanner) the mask table MT may beconnected to a short-stroke actuator only, or may be fixed. Mask MA andsubstrate W may be aligned using mask alignment marks M1, M2 andsubstrate alignment marks P1, P2. Although the substrate alignment marksas illustrated occupy dedicated target portions, they may be located inspaces between target portions (these are known as scribe-lane alignmentmarks). Similarly, in situations in which more than one die is providedon the mask MA, the mask alignment marks may be located between thedies.

The depicted apparatus could be used in at least one of the followingmodes:

1. In step mode, the mask table MT and the substrate table WTa/WTb arekept essentially stationary, while an entire pattern imparted to theradiation beam is projected onto a target portion C at one time (i.e. asingle static exposure).

The substrate table WTa/WTb is then shifted in the X and/or Y directionso that a different target portion C can be exposed. In step mode, themaximum size of the exposure field limits the size of the target portionC imaged in a single static exposure.

2. In scan mode, the mask table MT and the substrate table WTa/WTb arescanned synchronously while a pattern imparted to the radiation beam isprojected onto a target portion C (i.e. a single dynamic exposure). Thevelocity and direction of the substrate table WTa/WTb relative to themask table MT may be determined by the (de-)magnification and imagereversal characteristics of the projection system PS. In scan mode, themaximum size of the exposure field limits the width (in the non-scanningdirection) of the target portion in a single dynamic exposure, whereasthe length of the scanning motion determines the height (in the scanningdirection) of the target portion.

3. In another mode, the mask table MT is kept essentially stationaryholding a programmable patterning device, and the substrate tableWTa/WTb is moved or scanned while a pattern imparted to the radiationbeam is projected onto a target portion C. In this mode, generally apulsed radiation source is employed and the programmable patterningdevice is updated as required after each movement of the substrate tableWTa/WTb or in between successive radiation pulses during a scan. Thismode of operation can be readily applied to maskless lithography thatutilizes programmable patterning device, such as a programmable mirrorarray of a type as referred to above.

Combinations and/or variations on the above described modes of use orentirely different modes of use may also be employed.

Lithographic apparatus LA in this example is of a so-called dual stagetype which has two substrate tables WTa and WTb and two stations—anexposure station and a measurement station—between which the substratetables can be exchanged. While one substrate on one substrate table isbeing exposed at the exposure station EXP, another substrate can beloaded onto the other substrate table at the measurement station MEA sothat various preparatory steps may be carried out. The preparatory stepsmay include mapping the surface height of the substrate using a levelsensor LS and measuring the position of alignment marks on the substrateusing an alignment sensor AS. The alignment marks are arranged nominallyin a regular grid pattern. However, due to inaccuracies in creating themarks and also due to deformations of the substrate that occurthroughout its processing, the marks deviate from the ideal grid.Consequently, in addition to measuring position and orientation of thesubstrate, the alignment sensor in practice must measure in detail thepositions of many marks across the substrate area, if the apparatus LAis to print product features at the correct locations with very highaccuracy. The measurement of alignment marks is therefore verytime-consuming and the provision of two substrate tables enables asubstantial increase in the throughput of the apparatus. If the positionsensor IF is not capable of measuring the position of the substratetable while it is at the measurement station as well as at the exposurestation, a second position sensor may be provided to enable thepositions of the substrate table to be tracked at both stations.

The apparatus further includes a lithographic apparatus control unitLACU which controls all the movements and measurements of the variousactuators and sensors described. LACU also includes signal processingand data processing capacity to implement desired calculations relevantto the operation of the apparatus. In practice, control unit LACU willbe realized as a system of many sub-units, each handling the real-timedata acquisition, processing and control of a subsystem or componentwithin the apparatus. For example, one processing subsystem may bededicated to servo control of the substrate positioner PW. Separateunits may even handle coarse and fine actuators, or different axes.Another unit might be dedicated to the readout of the position sensorIF. Overall control of the apparatus may be controlled by a centralprocessing unit, communicating with these sub-systems processing units,with operators and with other apparatuses involved in the lithographicmanufacturing process.

FIG. 2 at 200 shows the lithographic apparatus LA in the context of anindustrial production facility for semiconductor products. Within thelithographic apparatus (or “litho tool” 200 for short), the measurementstation MEA is shown at 202 and the exposure station EXP is shown at204. The control unit LACU is shown at 206. Within the productionfacility, apparatus 200 forms part of a “litho cell” or “litho cluster”that contains also a coating apparatus 208 for applying photosensitiveresist and other coatings to substrate W for patterning by the apparatus200. At the output side of apparatus 200, a baking apparatus 210 anddeveloping apparatus 212 are provided for developing the exposed patterninto a physical resist pattern.

Once the pattern has been applied and developed, patterned substrates220 are transferred to other processing apparatuses such as areillustrated at 222, 224, 226. A wide range of processing steps isimplemented by various apparatuses in a typical manufacturing facility.For the sake of example, apparatus 222 in this embodiment is an etchingstation, and apparatus 224 performs a post-etch annealing step. Furtherphysical and/or chemical processing steps are applied in furtherapparatuses, 226, etc. Numerous types of operation can be required tomake a real device, such as deposition of material, modification ofsurface material characteristics (oxidation, doping, ion implantationetc.), chemical-mechanical polishing (CMP), and so forth. The apparatus226 may, in practice, represent a series of different processing stepsperformed in one or more apparatuses.

As is well known, the manufacture of semiconductor devices involves manyrepetitions of such processing, to build up device structures withappropriate materials and patterns, layer-by-layer on the substrate.Accordingly, substrates 230 arriving at the litho cluster may be newlyprepared substrates, or they may be substrates that have been processedpreviously in this cluster or in another apparatus entirely. Similarly,depending on the required processing, substrates 232 on leavingapparatus 226 may be returned for a subsequent patterning operation inthe same litho cluster, they may be destined for patterning operationsin a different cluster, or they may be finished products to be sent fordicing and packaging.

Each layer of the product structure requires a different set of processsteps, and the apparatuses 226 used at each layer may be completelydifferent in type. Further, even where the processing steps to beapplied by the apparatus 226 are nominally the same, in a largefacility, there may be several supposedly identical machines working inparallel to perform the step 226 on different substrates.

Small differences in set-up or faults between these machines can meanthat they influence different substrates in different ways. Even stepsthat are relatively common to each layer, such as etching (apparatus222) may be implemented by several etching apparatuses that arenominally identical but working in parallel to maximize throughput. Inpractice, moreover, different layers require different etch processes,for example chemical etches, plasma etches, according to the details ofthe material to be etched, and special requirements such as, forexample, anisotropic etching.

The previous and/or subsequent processes may be performed in otherlithography apparatuses, as just mentioned, and may even be performed indifferent types of lithography apparatus. For example, some layers inthe device manufacturing process which are very demanding in parameterssuch as resolution and overlay may be performed in a more advancedlithography tool than other layers that are less demanding. Thereforesome layers may be exposed in an immersion type lithography tool, whileothers are exposed in a ‘dry’ tool. Some layers may be exposed in a toolworking at DUV wavelengths, while others are exposed using EUVwavelength radiation.

Also shown in FIG. 2 is a metrology apparatus 240 which is provided formaking measurements of parameters of the products at desired stages inthe manufacturing process. A common example of a metrology station in amodern lithographic production facility is a scatterometer, for examplean angle-resolved scatterometer or a spectroscopic scatterometer, and itmay be applied to measure properties of the developed substrates at 220prior to etching in the apparatus 222. Using metrology apparatus 240, itmay be determined, for example, that important performance parameterssuch as overlay or critical dimension (CD) do not meet specifiedaccuracy requirements in the developed resist. Prior to the etchingstep, the opportunity exists to strip the developed resist and reprocessthe substrates 220 through the litho cluster. As is also well known, themetrology results 242 from the apparatus 240 can be used to maintainaccurate performance of the patterning operations in the litho cluster,by control unit LACU 206 making small adjustments over time, therebyminimizing the risk of products being made out-of-specification, andrequiring re-work. Of course, metrology apparatus 240 and/or othermetrology apparatuses (not shown) can be applied to measure propertiesof the processed substrates 232, 234, and incoming substrates 230.

FIG. 3 illustrates the steps to expose target portions (e.g. die) on asubstrate W in the dual stage apparatus of FIG. 1.

On the left hand side within a dotted box are steps performed at ameasurement station MEA, while the right hand side shows steps performedat the exposure station EXP. From time to time, one of the substratetables WTa, WTb will be at the exposure station, while the other is atthe measurement station, as described above. For the purposes of thisdescription, it is assumed that a substrate W has already been loadedinto the exposure station. At step 300, a new substrate W′ is loaded tothe apparatus by a mechanism not shown. These two substrates areprocessed in parallel in order to increase the throughput of thelithographic apparatus.

Referring initially to the newly-loaded substrate W′, this may be apreviously unprocessed substrate, prepared with a new photo resist forfirst time exposure in the apparatus. In general, however, thelithography process described will be merely one step in a series ofexposure and processing steps, so that substrate W′ has been throughthis apparatus and/or other lithography apparatuses, several timesalready, and may have subsequent processes to undergo as well.Particularly for the problem of improving overlay performance, the taskis to ensure that new patterns are applied in exactly the correctposition on a substrate that has already been subjected to one or morecycles of patterning and processing. These processing stepsprogressively introduce distortions in the substrate that must bemeasured and corrected for, to achieve satisfactory overlay performance.

At 302, alignment measurements using the substrate marks P1 etc. andsensors (not shown) are used to measure and record alignment of thesubstrate relative to substrate table WTa/WTb. In addition, severalalignment marks across the substrate W′ will be measured using alignmentsensor AS. These measurements are used in one embodiment to establish aso-called wafer grid, which maps very accurately the spatialdistribution of alignment marks across the substrate, including anydistortion relative to a nominal rectangular grid. In other words, themeasurements record positional deviations of points on the substrate,relative to their ideal location.

At step 304, a map of wafer height (Z) against X-Y position is measuredalso using the level sensor LS. The height map is used to achieveaccurate focusing of the exposed pattern. Again, the measurements recordpositional deviations of points on the substrate in the Z direction,relative to an ideal (flat) substrate.

When substrate W′ was loaded, recipe data 306 were received, definingthe exposures to be performed, and also properties of the wafer and thepatterns previously made and to be made upon it. Recipe data 306 mayalso include high-order alignment model parameters obtained fromprevious metrology measurements. To these recipe data are added themeasurements of wafer position, wafer grid and height map that were madeat 302, 304, so that a complete set of recipe data and measurement data308 can be passed to the exposure station EXP. The measurements ofalignment data for example comprise X and Y positions of alignmenttargets formed in a fixed or nominally fixed relationship to the productpatterns that are the product of the lithographic process. Thesealignment data, taken just before exposure, are combined andinterpolated to provide parameters of an alignment model. Theseparameters and the alignment model will be used during the exposureoperation to correct positions of patterns applied in the currentlithographic step. A conventional alignment model might comprise four,five or six parameters, together defining translation, rotation andscaling of the ‘ideal’ grid, in different dimensions. As describedfurther in US 2013230797A1, advanced models are known that use moreparameters.

In this regard, the present description refers primarily to so-called“interfield” substrate models, which describe positional deviations thatare characteristic of locations across the substrate. In a real process,it is common also to model “intrafield” variations that arecharacteristic of locations within each field (target portion C). Todetermine the final position of applying a pattern, the interfield modeland intrafield model can be combined in a well-known manner.

At 310, wafers W′ and W are swapped, so that the measured substrate W′takes on the role of the substrate W, to be exposed as discussedpreviously, entering the exposure station EXP. In the example apparatusof FIG. 1, this swapping is performed by exchanging the supports WTa andWTb within the apparatus, so that the substrates W, W′ remain accuratelyclamped and positioned on those supports, to preserve relative alignmentbetween the substrate tables and substrates themselves. The wafer W thathas actually been exposed is removed and the relevant support willreceive a new substrate (not shown) for being subjected to themeasurements. Accordingly, once the tables have been swapped,determining the relative position between projection system PS andsubstrate table WTb (formerly WTa) is all that is necessary to make useof the measurement information 302, 304 for the substrate W (formerlyW′) in control of the exposure steps. At step 312, reticle alignment isperformed using the mask alignment marks M1, M2. In steps 314, 316, 318,scanning motions and radiation pulses are applied at successive targetlocations across the substrate W, in order to complete the exposure of anumber of patterns.

By using the alignment data and height map, as obtained at the measuringstation, in the performance of the exposure steps, these patterns areaccurately aligned with respect to the desired locations, and, inparticular, with respect to features previously laid down on the samesubstrate. The exposed substrate, now labeled W″ is unloaded from theapparatus at step 320, to undergo etching or other processes, inaccordance with the exposed pattern.

FIGS. 4A to 4C illustrate the form of alignment information that can beused to correct for wafer grid distortion as measured by the alignmentsensor AL on alignment marks (targets) 400 in a previous layer on wafer(substrate) W. As shown in FIG. 4A, each target has a nominal position,defined usually in relation to a regular, rectangular grid 402 with axesX and Y. With reference to FIG. 4B, measurements of the real position404 of each target reveal deviations from the nominal grid. Thealignment marks may be provided within device areas of the substrate,and/or they may be provided in so-called “scribe lane” areas betweendevice areas.

With reference to FIG. 4C, the measured positions 404 of all the targetscan be processed numerically to set up a model of a distorted wafer grid406 for this particular wafer. This alignment model is used in thepatterning operation to control the position of the patterns applied tothe substrate. In the example illustrated, the straight lines of thenominal grid have become curves. For such a case, a higher-order(advanced) alignment model can be used instead of with a linearalignment model. It goes without saying that the distortions illustratedare exaggerated compared to the real situation.

Even when advanced alignment models are used, errors inevitably remainin the overlay performance of the lithographic apparatus. An individuallithographic apparatus may also perform differently than other onesprocessing the same substrate. In order that the substrates that areexposed by the lithographic apparatus are exposed correctly andconsistently, it is desirable to inspect exposed substrates to measureperformance parameters such as overlay errors between subsequent layers,line thicknesses, critical dimensions (CD), etc.

An inspection apparatus is therefore used to determine the properties ofthe substrates independently of the alignment sensors AS, and inparticular, how the properties of different substrates or of differentlayers of the same substrate vary from layer to layer. The inspectionapparatus (not shown in FIG. 3, but shown in FIG. 2 at 240) may beintegrated into the lithographic apparatus LA or the lithocell LC or maybe a stand-alone device. It may be a scatterometer, for example anangle-resolved scatterometer of the time described in published USpatent application US2006033921A1.

The inspection apparatus can also be used in an advanced process control(APC) system to calibrate individual lithographic apparatus and to allowdifferent tools to be used more interchangeably. Improvements to theapparatus's focus and overlay (layer-to-layer alignment) uniformity haverecently been achieved by the implementation of a stability module,leading to an optimized process window for a given feature size and chipapplication, enabling the continuation the creation of smaller, moreadvanced chips. The stability module in one embodiment automaticallyresets the system to a pre-defined baseline at regular intervals, forexample each day. More detail of lithography and metrology methodsincorporating the stability module can be found in US2012008127A1. Theknown example implements three main process control loops. The firstloop provides the local control of the lithography apparatus using thestability module and monitor wafers. The second (APC) loop is for localscanner control on-product (determining focus, dose, and overlay onproduct wafers).

The third control loop is to allow metrology integration into the second(APC) loop (e.g., for double patterning). All of these loops usemeasurements made by the inspection apparatus 240 in FIG. 2, in additionto the measurements made during the actual patterning operations of FIG.3.

As mentioned above, standard alignment models may have six parameters(effectively three per direction X & Y) and in addition there are moreadvanced alignment models. On the other hand, for the most demandingprocesses currently in use and under development, to achieve the desiredoverlay performance requires more detailed corrections of the wafergrid. While standard models might use fewer than ten parameters,advanced alignment models typically use more than 15 parameters, or morethan 30 parameters. Examples of advanced models are higher order waferalignment (HOWA) models, zone-alignment (ZA) and radial basis function(RBF) based alignment models. HOWA is a published technique based onsecond, third and higher order polynomial functions. Zone alignment isdescribed for example in Huang et al, “Overlay improvement by zonealignment strategy”, Proc. SPIE 6922, 69221G (2008). RBF modeling isdescribed in published US patent application 2012/0218533. The advancedmodels generate a complex description of the wafer grid that iscorrected for, during the exposure of the target layer. RBF and latestversions of HOWA provide particularly complex descriptions based on tensof parameters. This implies a great many measurements are required toobtain a wafer grid with sufficient detail.

The inventors have observed that, in some processes, the alignment andoverlay error radically increases in magnitude towards the edge of thewafer. FIG. 5 shows plots 502 and 504 of uncorrected overlay errors intwo different example products. The arrows indicate the direction ofalignment errors. The direction of error may be either inwards oroutwards, depending on the root cause and/or measurement convention(e.g. upper layer to lower layer alignment or vice versa). The edgeeffect is clearly visible in FIG. 5. Moreover, plots 502′ and 504′ showresidual errors (“residuals”) for the same products after fitting withan interfield polynomial model and an intrafield polynomial model. (Theresiduals are not necessarily on the same scale as the original errors).That is to say, the residuals in plots 502′ 504′ represent errors thatremain uncorrected even after application of modern high-ordercorrection models. At the edges of the wafers the magnitude of thearrows is larger and the direction can point radially inwards 502 orradially outwards 504. Thus it can be seen from the residuals thatfitting with a 5th order interfield polynomial is not sufficient forcapturing the edge effect. Similarly, other known advanced high-orderalignment models cannot adequately describe the edge effect.

FIG. 6 illustrates some possible root causes and effect on overlay ofwafer edge effects. Most likely the edge disturbances in the illustratedexamples are caused by a curvature in an electrical field during dryetching. In this type of etching, ions are moved under the electricfield to bombard the material to be etched. A cross-section of the waferand etching environment are shown in FIG. 6. As shown in FIG. 6 (a),both the etch chamber 602 and wafer 604 are finite in size.Iso-potential lines 606 (lines of equal electric potential) will not beparallel to the wafer surface close to the edge, causing ions 608 tobombard the surface in a non-perpendicular direction (see detail at(b)). The iso-potential lines in the two-dimensional cross-section stemfrom two-dimensional iso-potential surfaces in the three-dimensionalscene. The etched multi-layered structures 610 shown at (c) are slanted,and thus not neatly aligned to the underlying layer structures 612.

The inventors have appreciated that current alignment models are notable to accurately describe this effect. Known automatic process control(APC) systems will not be able to detect such edge disturbances,therefore it is not will not raise appropriate alarms, nor adequatelycorrect for these edge disturbances. As a consequence, the ability toimprove yield of edge dies using APC systems is adversely affected bythis edge effect.

A simple solution would be to increase the maximal fit order of currentstate of the art polynomial models. But since none of those high-orderbasis functions do correlate particularly well with the edgedisturbance, many additional degrees of freedom (for example10^(th)-order polynomials) would be needed in order to obtain sufficientaccuracy. As a consequence such models require many more measurementpoints and/or will be much more sensitive to noise. Furthermore, thoseextra basis functions and coefficients do not offer extra insight intothe disturbance root cause.

The inventors have recognized that the existing models for alignment andoverlay can be extended in a more efficient manner to provide methods ofcalculating alignment model parameters to overcome the problemsdiscussed above. In such extended alignment and overlay models, thedisturbance represented by the model may be a positional deviation. Theextended models are also envisioned to be applicable to otherlithographic patterning characteristics such as, but not limited to,critical dimension (CD), focus and sidewall angle (SWA), because thesubstrate-edge-related disturbances for these characteristics have asimilar root cause. Therefore the present disclosure makes reference tosubstrate models, which include alignment models, CD models, focusmodels and SWA models. Substrate models are used to model disturbancesintroduced in lithographic processing, and can represent the“fingerprint” or pattern of disturbance of a particular lithographicprocess or process step across a substrate. These disturbances include,but are not limited to alignment (positional deviations in the plane ofthe substrate), height deviations (normal to the plane), CD, focus andSWA.

In the proposed extended models, the interfield overlay disturbances maybe described in terms of their radial & tangential (R&T) components asseen from the wafer center, rather than the traditional Cartesian (X&Y)orientations in which the measurements are delivered.

A semiconductor wafer is generally circular, although it may have asmall flattened section or notch (seen in FIG. 4, for example) foralignment. If R is the distance from the center of the wafer to the edgeand r is the distance of a point from the center of the wafer, then lett=R−r be the distance from the wafer edge for any given exposure field.More generally, the principles of the present disclosure may be appliedin processing rectangular substrates, or other non-circular substrates.Depending on the substrate shape, and on the processing effects whichare to be modeled, a different coordinate system may be adopted toexpress the distance from the substrate edge.

According to principles of the present disclosure, one or more specificedge-related basis functions are added to the interfield substrate modelthat are designed to be more capable of describing the edge effect.Substrate model parameters are thus calculated using a combination ofbasis functions, the basis functions including at least one edge basisfunction related to a substrate edge.

Let u(t) be the edge basis function (typically having units ofnanometers). Various functions can be considered as suitable for use asedge basis functions related to a substrate edge. An exponential decayfunction may be defined:u(t)=C·2 ^(−t/λ)with λ being a half-life decay distance or decay range parameter and Cbeing the amplitude at the edge.

Another example is a rational function:u(t)=C₁ /t+C ₂ /t ²with C₁ and C₂ being shape constants.

When using a rational function as a basis function, as in the secondexample above, care should be taken to avoid “divide-by-zero” errors.Using the simple formula above, u(t) will be infinity when “t”approaches zero. In a practical implementation, therefore, somemodification of the formula is used to avoid excessive values, and toavoid computational error conditions. In one such implementation we usethe terms of the form C/(t+δ) in place of C/t, where δ is a small offseteffective to avoid dividing by zero at the wafer edge. (Alternatively,and equivalently, one can calculate t by reference to a radius R that isslightly larger than the true radius of the substrate.) Alternatively,one could apply a rule whereby the rational function is used only tovalues of “t” larger than a minimum value δ larger than zero. A varietyof measures can be envisaged.

In another example, δ may be used as one of the variable parameters ofthe edge effect model. For example, a functionu(t)=C ₁/(t+δ)can be envisaged, with C1 and δ in the role of shape constants. Afurther term

It will be noted that these example edge basis functions are basisfunctions having one or more contributions expressed in terms ofdistance from the wafer edge, t. In the exponential decay functions, oneor more terms have the distance from the edge as exponent. In theexample rational functions, one or more terms have the distance from theedge in a denominator. These forms can be combined and/or other edgebasis functions may be used. A feature of these examples is that theireffect can be limited to an arbitrarily narrow edge area of thesubstrate, i.e., to a surface area of the substrate that is bounded bythe substrate's perimeter and that has a radial width much smaller thanthe radius of the substrate. Accordingly, the edge basis functionsenable to take into account the spatial dependence of a specific one ofthe disturbances related to only an area of the substrate near an edgeof the substrate. In this way it does not disrupt the definition andfitting of the interfield model across the substrate as a whole. It willalso be noted that each example introduces only two additional degreesof freedom to the model. Consequently the additional computationalcomplexity is minimized, and the additional measurement burden isavoided.

The example edge basis functions given above also have the property thatthey are functions whose value decreases monotonically from thesubstrate edge. As known, a mathematical function F is called“monotonically decreasing” if, whenever p≤q, then F(p)≥F(q). Functionshaving such a property can be used to model the typical effects arisingat the edge of a substrate in processing situations such as the etchingoperation of FIG. 6.

Distance from the edge is defined in the above examples as radialdistance, relative to a circular substrate. The appropriate definitionof distance from the edge in a given situation can be determined fromthe geometry of the substrate and the nature of the edge-relatedprocessing effects.

FIG. 7 illustrates graphs of radial alignment error versus distance fromthe wafer edge, compared with fitted models having parameters calculatedin accordance with embodiments of the present disclosure. FIG. 7 showshow these edge basis functions (when used together with conventionalpolynomial basis functions) are capable of describing the average radialtrend without straying beyond the 95% confidence interval (CI) shown bythe solid non-smooth lines. 702 is a graph showing application of anexponential decay function with dots for the data points. 704 is a graphshowing application of a rational function. The rational function isshown as a solid line, with a dashed line for the data points. In FIG.7, the vertical axis is the radial alignment error AE (in arbitraryunits) and the horizontal axis is respectively distance from the waferedge, t, (in mm) or normalized with respect to R, i.e. t/R. The fit lineis shown as a smooth solid line in both graphs. This good fitillustrates that systematic edge effects are captured with minimalcomputational expense.

The exponential decay and rational edge basis functions described abovehave only two (additional) degrees of freedom. The result is an improvedalignment performance, without significantly changing the noisesuppression of the fit. This makes the extended substrate models verywell suited for use cases that have a sparse sampling layout, such aswafer alignment and inline metrology.

The extended substrate model for overlay is able to capture the slantedetch-induced alignment error close to the wafer edge. Such a model canbe implemented in a controller application, such as used in thecontroller LACU of FIG. 2 and/or any advanced process controlapplication. The lithographic system is thus able to detect thesedisturbances, and if desired correct for them to the best of theavailable lithographic apparatus actuator capabilities. Alarm signalsand other diagnostic information can also be derived from theedge-related parameters, which would not be available in a model whereedge effects are not specifically recognized.

Optionally, the extended substrate model may represent disturbances in aselected direction in the plane of the substrate. For example, for analignment model the extended model may be used for only one of the twooverlay directions X and Y, or may represent each in a separate model.This may be applicable in lithographic layers where X-overlay andY-overlay are measured to a different underlying layer. Only one of thelayers might be susceptible to an edge effect. The extended model may beselectively enabled or disabled simply, to suit different situations. Todisable the extended model, the edge basis functions can simply beomitted from the calculations, or the edge-related related parameterscan be substituted with zero (or whatever value is appropriate toproduce zero contribution in the overall model).

FIG. 8 illustrates a method of controlling lithographic processing usingthe techniques described above. The method is suitable for use in theproduction facility illustrated in FIG. 2 and the measurement andexposure process illustrated in FIG. 3.

In step 802, an extended high-order substrate model is defined using acombination of basis functions. The basis functions include at least oneedge basis function for representing effects related to a substrateedge.

In step 804 offline metrology measurements of structures on substrates(220 in FIG. 2) that have been subjected to lithographic processing areperformed using the metrology tool 240 of FIG. 2.

In step 806, the processor in the metrology tool receives themeasurements and calculates high-order substrate model parameters 808 byfitting the measurements to the extended model defined in step 802.Standard regression techniques may be used. The edge basis function maybe expressed in terms of distance from a substrate edge. The edge basisfunction may comprise a radial basis function. The edge basis functionmay comprise for example an exponential decay function or a rationalfunction. The edge basis function may comprise both an exponential decayfunction and a rational function. However, since it is generallydesirable to minimize the number of degrees of freedom down, using justone type of edge basis function is preferred. For an exponential decayedge basis function, the substrate model parameters may comprise anamplitude parameter of the exponential decay function and a decay rangeparameter of the exponential decay function. For a rational edge basisfunction, the substrate model parameters may comprise at least two shapeconstant parameters of the rational function. In either case, both theamplitude of the modeled edge effect and the degree to which it islocalized at the substrate edge can be modeled very simply.

Steps 810 to 820 describe control of the lithographic processing of asubsequent substrate using the high-order substrate model parameterscalculated in steps 802 to 808. Steps 810 to 820 are implemented inlithographic processing by the lithographic apparatus 200 shown in FIG.2. The high-order substrate model parameters 808 may be passed to thelithographic apparatus 200, at 242 in FIG. 2. The high-order substratemodel parameters may be included in the recipe 306 illustrated in FIG.3.

At step 810, a low-order substrate model is defined, such as aconventional 6PAR model. At step 812 the measurement station MEA (202 inFIG. 2) performs inline measurements of structures on a subsequentsubstrate. A subsequent substrate is one that is subject to lithographicprocessing after the previous lithographic processing of the substrateson which the offline measurements 804 have been done. The inlinemeasurements have a lower number of measurement points per substrate(i.e. density) than the offline metrology measurements.

At step 814, the processor in the lithography apparatus controller (LACU206 in FIG. 2) calculates low-order substrate model parameters 816 byfitting the inline measurements to the low-order model defined in step810. Standard regression techniques may be used.

In step 818 the high-order parameters 808 are combined with thelow-order model parameters 816 in a conventional manner to define ahigh-order substrate model specific to the substrate being patterned.

At step 820, the processor in the lithography apparatus controller (LACU206 in FIG. 2) controls lithographic processing of the subsequentsubstrate using the calculated high-order substrate model parameters 808in combination 818 with the low-order substrate model parameters 816.This corresponds to the steps 314, 316 and 318 of FIG. 3.

There are several benefits that may be attained using the extendedsubstrate model according to the principles disclosed herein. Benefitsfor the device manufacturer are yield improvement and more efficientpreventive maintenance of equipment such as etchers.

Further advantages can be obtained when one recognizes that the edgeeffect parameters of the model effectively separate process and scannercontributions. The separation of the edge-related parameters in thesubstrate model provides an easy way for the system operator to enableor disable the correction of edge-related effects. The separation isuseful because not all identified effects can be or should be corrected.The substrate model may represent disturbances in a selected direction,such as X or Y, by selectively enabling or disabling the correction ofedge-related effects in each of the X and Y directions.

Defining a correction may be regarded as a permanent solution to theissues revealed by the analysis, or it may be that servicing orreplacement of a responsible apparatus is required. Correction may beuseful as a temporary measure until the responsible processing apparatuscan be recalibrated or repaired. If the error is not correctablesufficiently, the apparatus in question may be omitted from processing(or reassigned to less critical operations).

In addition, it should be noted that to correct some types of errormight make a performance parameter such as overlay worse, not better.This is because disturbances that remain consistent from layer to layerintroduces no overlay error at all, whereas to identify such deviationsand attempt to correct them in subsequent layers would introduceoverlay. In additional to overlay between features in different layersof product features on a substrate, overlay may be defined as errors inrelative placement of features within one layer, when performed byseparate patterning steps. One example of such “multiple patterning”processes is a LELE (Litho-Etch-Litho-Etch) double patterning process.In such a case in which the overlay is measured between the secondlithographic layer with respect to the first etched layer, both layersultimately forming a single layer of (high density) product features inthe functional device.

Having identified when certain edge effects should not be corrected, asimple indication of this can be stored as part of the alignment“recipe”. The edge-related parameters can be ignored (set to zero) whencalculating the corrections for each product unit (substrate).

The steps of the methods described above can be automated within anygeneral purpose data processing hardware (computer), so long as it hasaccess to the measurement data. The apparatus may be integrated withexisting processors such as the lithography apparatus control unit LACUshown in FIG. 2 or an overall process control system. The hardware canbe remote from the processing apparatus, even being located in adifferent country. Components of a suitable data processing apparatus(DPA) are shown in FIG. 9. The apparatus may be arranged for loading acomputer program product comprising computer executable code. This mayenable the computer assembly, when the computer program product isdownloaded, to implement the functions of the PCA apparatus and/or RCAapparatus as described above.

With reference to FIG. 9, memory 929 connected to processor 927 maycomprise a number of memory components like a hard disk 961, Read OnlyMemory (ROM) 962, Electrically Erasable Programmable Read Only Memory(EEPROM) 963 and Random Access Memory (RAM) 964. Not all aforementionedmemory components need to be present. Furthermore, it is not essentialthat aforementioned memory components are physically in close proximityto the processor 927 or to each other. They may be located at a distanceaway

The processor 927 may also be connected to some kind of user interface,for instance a keyboard 965 or a mouse 966. A touch screen, track ball,speech converter or other interfaces that are known to persons skilledin the art may also be used.

The processor 927 may be connected to a reading unit 967, which isarranged to read data, e.g. in the form of computer executable code,from and under some circumstances store data on a data carrier, like afloppy disc 968 or a CDROM 969. Also DVD's or other data carriers knownto persons skilled in the art may be used.

The processor 927 may also be connected to a printer 970 to print outoutput data on paper as well as to a display 971, for instance a monitoror LCD (Liquid Crystal Display), of any other type of display known to aperson skilled in the art.

The processor 927 may be connected to a communications network 972, forinstance a public switched telephone network (PSTN), a local areanetwork (LAN), a wide area network (WAN) etc. by means oftransmitters/receivers 973 responsible for input/output (I/O). Theprocessor 927 may be arranged to communicate with other communicationsystems via the communications network 972. In an embodiment of theinvention external computers (not shown), for instance personalcomputers of operators, can log into the processor 927 via thecommunications network 972.

The processor 927 may be implemented as an independent system or as anumber of processing units that operate in parallel, wherein eachprocessing unit is arranged to execute sub-tasks of a larger program.The processing units may also be divided in one or more main processingunits with several sub-processing units. Some processing units of theprocessor 927 may even be located a distance away of the otherprocessing units and communicate via communications network 972.Connections between modules can be made wired or wireless.

The computer system can be any signal processing system with analogueand/or digital and/or software technology arranged to perform thefunctions discussed here.

Although specific reference may be made in this text to the use oflithographic apparatus in the manufacture of ICs, it should beunderstood that the lithographic apparatus described herein may haveother applications, such as the manufacture of integrated opticalsystems, guidance and detection patterns for magnetic domain memories,flat-panel displays, liquid-crystal displays (LCDs), thin film magneticheads, etc. As already mentioned, the invention may be applied inindustrial processing applications quite separate from lithography.Examples might be in production of optical components, automotivemanufacture, construction—any number of applications where object dataexists in the form of measurements made with a certain spatialdistribution over the product. As in the example of lithography.Although specific reference may have been made above to the use ofembodiments of the invention in the context of optical lithography, itwill be appreciated that the invention may be used in other types oflithography, for example imprint lithography, and where the contextallows, is not limited to optical lithography. In imprint lithography atopography in a patterning device defines the pattern created on asubstrate. The topography of the patterning device may be pressed into alayer of resist supplied to the substrate whereupon the resist is curedby applying electromagnetic radiation, heat, pressure or a combinationthereof. The patterning device is moved out of the resist leaving apattern in it after the resist is cured.

The terms “radiation” and “beam” used herein encompass all types ofelectromagnetic radiation, including ultraviolet (UV) radiation (e.g.having a wavelength of or about 365, 248, 193, 157 or 126 nm) andextreme ultra-violet (EUV) radiation (e.g. having a wavelength in therange of 5-20 nm), as well as particle beams, such as ion beams orelectron beams.

The descriptions above are intended to be illustrative, not limiting.Thus, it will be apparent to one skilled in the art that modificationsmay be made to the invention as described without departing from thespirit and scope of the claims appended hereto. In addition, it shouldbe appreciated that structural features or method steps shown ordescribed in any one embodiment herein can be used in other embodimentsas well.

Aspects of the concept of the invention are summarized below in someclauses:

-   1. A method of calculating a substrate model for use in controlling    a lithographic process, the method comprising:    -   defining a substrate model for representing disturbances of        features on substrate to which patterns are to be applied by the        lithographic process, the substrate model being defined as a        combination of predefined basis functions;    -   receiving measurements of structures on at least one substrate;        and    -   calculating substrate model parameters using the measurements        and the basis functions,

wherein the basis functions include at least one edge basis function forrepresenting effects related to a substrate edge.

-   2. The method of clause 1, wherein the edge basis function decreases    monotonically with increasing distance from a substrate edge.-   3. The method of clause 1 or 2, wherein the substrate is of    generally circular form, and the edge basis function comprises a    radial basis function.-   4. The method of clause 1 or 2, wherein the substrate model    parameters comprise two parameters related to said edge basis    function or edge basis functions.-   5. The method of clause 1 or 2, wherein the substrate model    represents disturbances in a selected direction in the plane of the    substrate.-   6. An apparatus for calculating a substrate model for use in    controlling a lithographic process, the apparatus comprising a data    processing apparatus programmed to perform the steps of:    -   defining a substrate model for representing disturbances of        features on substrate to which patterns are to be applied by the        lithographic process, the substrate model being defined as a        combination of predefined basis functions;    -   receiving measurements of structures on at least one substrate;        and    -   calculating substrate model parameters using the measurements        and the basis functions,

wherein the basis functions include at least one edge basis function forrepresenting effects related to a substrate edge.

-   7. The apparatus of clause 6, wherein the edge basis function    decreases monotonically with increasing distance from a substrate    edge.-   8. The apparatus of clause 6 or 7, wherein the substrate is of    generally circular form, and the edge basis function comprises a    radial basis function.-   9. The apparatus of clause 8, wherein the substrate model parameters    comprise two parameters related to said edge basis function or edge    basis functions.-   10. The apparatus of clause 6 or 7, wherein the substrate model is    defined for representing disturbances in a selected direction in the    plane of the substrate.-   11. The apparatus of any of clause 6 or 7, wherein the edge basis    function can be selectively enabled or disabled.-   12. A method of controlling lithographic processing in which    patterns are applied to substrates, the method comprising:    -   receiving first measurements of structures on substrates that        have been subjected to lithographic processing;    -   using said first measurements of disturbances to calculate first        substrate model parameters using a method as specified in any of        clauses 1 to 5; and    -   controlling lithographic processing of a subsequent substrate        using said calculated first substrate model parameters.-   13. The method of controlling lithographic processing of clause 12,    wherein the step of controlling lithographic processing of the    subsequent substrate comprises:    -   receiving second measurements of disturbances on the subsequent        substrate, the second measurements having a lower number per        substrate than the first measurements;    -   using said second measurements to calculate second substrate        model parameters of the subsequent substrate using a model        having a lower order than a model used to calculate the first        substrate model parameters; and    -   controlling lithographic processing of the subsequent substrate        using said calculated first substrate model parameters in        combination with said second substrate model parameters.-   14. The method of clause 12 or 13 wherein the step of controlling    lithographic processing includes selectively enabling or disabling    use of substrate model parameters related to the edge basis    function.-   15. An apparatus for controlling lithographic processing in which    substrates are subjected to lithographic processing, the apparatus    comprising a data processing apparatus programmed to perform the    steps of:    -   receiving first measurements of disturbances on substrates that        have been subjected to lithographic processing;    -   using said first measurements to calculate first substrate model        parameters using a method as specified in any of clauses 1 to 5;        and    -   controlling lithographic processing of a subsequent substrate        using said calculated first substrate model parameters.-   16. The apparatus of clause 15, wherein for controlling lithographic    processing of the subsequent substrate the data processing apparatus    is programmed to perform the steps of:    -   receiving second measurements of disturbances on the subsequent        substrate, the second measurements having a lower number per        substrate than the first measurements;    -   using said second measurements to calculate second substrate        model parameters of the subsequent substrate using a model        having a lower order than a model used to calculate the first        substrate model parameters; and    -   controlling lithographic processing of the subsequent substrate        using said calculated first substrate model parameters in        combination with said second substrate model parameters.-   17. The apparatus of clause 15 or 16 wherein for controlling    lithographic processing the data processing apparatus is arranged to    selectively enable or disable use of substrate model parameters    related to said the edge basis function.-   18. A computer program product comprising machine readable    instructions for causing a general purpose data processing apparatus    to perform the steps of a method as specified in any of clauses 1 to    5 or 12 to 14.-   19. A computer program product as specified in clause 18 further    comprising machine readable instructions for causing said data    processing apparatus to implement an apparatus as specified in any    of clauses 6 to 11 or 15 to 17.

The invention claimed is:
 1. A method of controlling a lithographicapparatus configured for processing a substrate, the processingcomprising applying a pattern to the substrate, and the methodcomprising: using a mathematical substrate model for representingdisturbances of first features on the substrate, the mathematicalsubstrate model including a predefined edge basis function and at leastone substrate model parameter, wherein the edge basis function isrepresentative of spatial dependence of a specific one of thedisturbances related to only an area of the substrate near an edge ofthe substrate; receiving measurements of second features, similar to thefirst features, on at least one previous substrate, other than or inaddition to the substrate, that has been subjected to the processing;using the measurements and the edge basis function to determine a valueof the substrate model parameter; and controlling the lithographicapparatus in accordance with the mathematical substrate model using thevalue of the substrate model parameter.
 2. The method of claim 1,wherein the edge basis function decreases monotonically with increasingdistance from the edge.
 3. The method of claim 1, wherein the edge ofthe substrate is generally circular; and the edge basis function is aradial basis function.
 4. The method of claim 1, comprising selectivelyenabling or selectively disabling including the edge basis function intothe mathematical substrate model.
 5. The method of claim 1, wherein thesubstrate model parameter comprises at least two parameters related tothe edge basis function.
 6. The method of claim 1, wherein the substratemodel represents disturbances in a selected direction in the plane ofthe substrate.
 7. The method of claim 1, wherein the mathematicalsubstrate model further comprises one or more substrate basis functionshaving at least five substrate basis function parameters, wherein theone or more substrate basis functions are representative of the specificone of the disturbances related to an area of the substrate including anarea outside of the area of the substrate near the edge of the substrateas well including the area of the substrate near the edge of thesubstrate, and wherein the substrate model parameter is at least oneedge basis function parameter of the predefined edge basis function. 8.A non-transitory computer program product comprising machine-readableinstructions for being executed by a data processing apparatus, whereinthe machine-readable instructions comprise; instructions configured forinvoking a mathematical substrate model for representing disturbances offirst features on a substrate for processing by applying a pattern tothe substrate, the mathematical substrate model including a predefinededge basis function and at least one substrate model parameter, whereinthe edge basis function is representative of spatial dependence of aspecific one of the disturbances related to only an area of thesubstrate near an edge of the substrate; instructions configured forreceiving measurements of second features, similar to the firstfeatures, on at least one previous substrate, other than or in additionto the substrate, that has been subjected to the processing;instructions configured for using the measurements and the edge basisfunction to determine a value of the substrate model parameter; andinstructions configured for controlling a lithographic apparatus inaccordance with the mathematical substrate model using the value of thesubstrate model parameter.
 9. The computer program product of claim 8,wherein the edge basis function decreases monotonically with increasingdistance from the edge.
 10. The computer program product of claim 8,wherein the edge of the substrate is generally circular; and the edgebasis function is a radial basis function.
 11. The computer programproduct of claim 8, comprising instructions configured to selectivelyenable or selectively disable including the edge basis function into themathematical substrate model.
 12. The computer program product of claim8, wherein the substrate model parameter comprises at least twoparameters related to the edge basis function.
 13. The computer programproduct of claim 8, wherein the substrate model represents disturbancesin a selected direction in the plane of the substrate.
 14. The computerprogram product of claim 8, wherein the mathematical substrate modelfurther comprises one or more substrate basis functions having at leastfive substrate basis function parameters, wherein the one or moresubstrate basis functions are representative of the specific one of thedisturbances related to an area of the substrate including an areaoutside of the area of the substrate near the edge of the substrate aswell including the area of the substrate near the edge of the substrate,and wherein the substrate model parameter is at least one edge basisfunction parameter of the predefined edge basis function.
 15. Anapparatus to control lithographic processing of substrates, theapparatus comprising a data processing apparatus programmed to: invoke amathematical substrate model for representing disturbances of firstfeatures on a substrate for lithographic processing, the mathematicalsubstrate model including a predefined edge basis function and at leastone substrate model parameter, wherein the edge basis function isrepresentative of spatial dependence of a specific one of thedisturbances related to only an area of the substrate near an edge ofthe substrate; receive measurements of second features, similar to thefirst features, on at least one previous substrate, other than or inaddition to the substrate, that has been subjected to the lithographicprocessing; use the measurements and the edge basis function todetermine a value of the substrate model parameter; and generate asignal for control of a lithographic apparatus in accordance with themathematical substrate model using the value of the substrate modelparameter.
 16. The apparatus of claim 15, wherein the edge basisfunction decreases monotonically with increasing distance from the edge.17. The apparatus of claim 15, wherein the edge of the substrate isgenerally circular; and the edge basis function is a radial basisfunction.
 18. The apparatus of claim 15, programmed to selectivelyenable or selectively disable including the edge basis function into themathematical substrate model.
 19. The apparatus of claim 15, wherein thesubstrate model parameter comprises at least two parameters related tothe edge basis function.
 20. The apparatus of claim 15, wherein thesubstrate model represents disturbances in a selected direction in theplane of the substrate.
 21. The apparatus of claim 15, comprising ametrology tool configured to generate the measurements of secondfeatures.
 22. The apparatus of claim 15, comprising the lithographicapparatus.